The primary objective is to utilize the 'molecular clock hypothesis' to develop mathematical models that will allow us to study how cancers grow and spread. Human cancer growth cannot be directly observed and the overall goal is to develop an approach that can retrospectively reconstruct tumor progression by "reading" the ancestry surreptitiously written within genomes by replication errors. Sequences are commonly used to reconstruct the genealogy of species and individuals, and we propose to translate this general molecular phylogeny approach to human cancers. We will use DMA methylation data, an epigenetic modification of DNA that is replicated at cell division. As direct calculation can be either impractical or infeasible, we propose to use rejection algorithms, a simulation-based approach. This general framework will allow us to estimate the age of a tumor, the age of a metastasis, the methylation error rate, and whether the metastasis is derived from a population of cells from the primary cancer. Our aims are motivated by ongoing studies at the Morris Comprehensive Cancer Center at the University of Southern California. Specifically, we propose to: 1. Develop methods that will allow us to estimate parameters characterizing the growth of cancer using 5' to 3' DNA methylation patterns. The models will address the following biological problems: a. To test for the existence of cancer stem cells based on the types of ancestral trees inferred from the methylation patterns b. To evaluate tumor heterogeneity c. To estimate tumor age and the rate of methylation errors 2. Extend models developed in Aim 1 to study the spread of cancer. The goal will be to compare two cell populations (primary tumor and metastasis) and determine if they are the same age, or if one is younger and derived from the other. 3. Extend the model in Aims 1 and 2 to allow the probability of methylation at each CpG site to depend on the methylation status of neighboring CpGs and evaluate its effect on the biological questions of interest. 4. Apply the methods to DNA methylation patterns observed in primary tumors of the colon and distant metastasis in humans and in mice. [unreadable] [unreadable] [unreadable] [unreadable]